I learned something yesterday. Euclid gave a beautiful example of an infinitesimal, a quantity less than any positive quantity. It's the horn angle of a circle on a line. That is, if you draw a circle tangent to a line, the circumference of the circle makes an angle with the line that is less than the angle made by any other line through that point in the direction of the circle. Of course today we'd say the horn angle is zero, since that's the limit. But it's interesting that Euclid recognized the idea of a quantity that was strictly smaller than any other positive quantity.

Of course today we'd say the horn angle is zero, since that's the limit.

We wouldn't though. Only recently Euclid's idea about the horn angle was vindicated. In algebraic geometry today, infinitesimals are allowed. And stuff like the Horn angles become easier to study with infinitesimals. The 1800's might have tried to shun the infinitesimal, in the 1900's we tried to get it to come back to us.

We wouldn't though. Only recently Euclid's idea about the horn angle was vindicated. In algebraic geometry today, infinitesimals are allowed. And stuff like the Horn angles become easier to study with infinitesimals. The 1800's might have tried to shun the infinitesimal, in the 1900's we tried to get it to come back to us.

I've heard of smooth infinitesimal analysis but I don't know anything about it.

In basic analysis the angle between two curves is the limit of their tangents, yes? Which is 0 in this case. Appreciate any clarity on this point.

I've heard of smooth infinitesimal analysis but I don't know anything about it.

In basic analysis the angle between two curves is the limit of their tangents, yes? Which is 0 in this case. Appreciate any clarity on this point.

Yeah, that's fine. The angle is definitely 0 in basic analysis and differential geometry. I wasn't contradicting you or anything. I was just trying to say that Euclid's idea of horn angles was very genius, because now we came up with similar things in very advanced topics like algebraic geometry.

No. It is an arbitrarily small distance between two points. It is the fundamental concept behind calculus.

This is ill-informed nonsense. Standard analysis and thus calculus uses only the finite - finitely large and finitely small.

Non-standard analysis makes use of infinitesimals to simplify some concepts and is closely related to how the theory was first developed, but infinitesimals are not required and certainly not fundamental.